nLab
cancellative midpoint algebra

Contents

Contents

Idea

The idea of a cancellative midpoint algebra comes from Peter Freyd.

Definition

A cancellative midpoint algebra is a midpoint algebra (M,|)(M,\vert) that satisfies the cancellative property:

  • for all aa, bb, and cc in MM, if a|b=a|ca \vert b = a \vert c, then b=cb = c

Examples

The rational numbers, real numbers, and the complex numbers with a|ba+b2a \vert b \coloneqq \frac{a + b}{2} are examples of cancellative midpoint algebras.

The trivial group with a|b=aba \vert b = a \cdot b is a cancellative midpoint algebra.

References

  • Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

Last revised on June 1, 2021 at 14:30:09. See the history of this page for a list of all contributions to it.